In September 2000, at the Centre de Recerca Matematica in Barcelona, we pre sented a 30-
hour Advanced Course on Algebraic Quantum Groups. After the course, we expanded and …

The first book devoted to cluster algebras, this work contains chapters on Poisson geometry
and Schubert varieties; an introduction to cluster algebras and their main properties; and …

We introduce a Poisson variety compatible with a cluster algebra structure and a compatible
toric action on this variety. We study Poisson and topological properties of the union of …

The text is devoted to the study of algebras of functions on quantum groups. The book
includes the theory of Poisson-Lie algebras (quasi-classical version of algebras of functions …

We introduce a new class of algebras called Poisson orders. This class includes the
symplectic reflection algebras of Etingof and Ginzburg, many quantum groups at roots of …

We generalize the classical study of (generalized) Lax pairs, the related\mathcal O-
operators and the (modified) classical Yang-Baxter equation by introducing the concepts of …

We prove that a Noetherian Hopf algebra of finite global dimension possesses further
attractive homological properties, at least when it satisfies a polynomial identity. This applies …

All algebras in a very large, axiomatically defined class of quantum nilpotent algebras are
proved to possess quantum cluster algebra structures under mild conditions. Furthermore, it …

We study prime and primitive ideals in a unified setting applicable to quantizations (at
nonroots of unity) of $ n\times n $ matrices, of Weyl algebras, and of Euclidean and …

It is proved that every prime factor algebra of the coordinate ring ${\mathcal {O} _q}({M_n}(k))
$ of quantum $ n\times n $ matrices over a field k is an integral domain (albeit not …